Information Fusion in Signal and Image Processing: Major Probabilistic and Non-Probabilistic Numerical Approaches

Preface 11
Isabelle BLOCH

Chapter 1. Definitions 13
Isabelle BLOCH and Henri MAÎTRE

1.1. Introduction 13

1.2. Choosing a definition 13

1.3. General characteristics of the data 16

1.4. Numerical/symbolic 19

1.4.1. Data and information 19

1.4.2. Processes 19

1.4.3. Representations 20

1.5. Fusion systems 20

1.6. Fusion in signal and image processing and fusion in other fields 22

1.7. Bibliography 23

Chapter 2. Fusion in Signal Processing 25
Jean-Pierre LE CADRE, Vincent NIMIER and Roger REYNAUD

2.1. Introduction 25

2.2. Objectives of fusion in signal processing 27

2.2.1. Estimation and calculation of a law a posteriori 28

2.2.2. Discriminating between several hypotheses and identifying 31

2.2.3. Controlling and supervising a data fusion chain 34

2.3. Problems and specificities of fusion in signal processing 37

2.3.1. Dynamic control 37

2.3.2. Quality of the information 42

2.3.3. Representativeness and accuracy of learning and a priori information 43

2.4. Bibliography 43

Chapter 3. Fusion in Image Processing 47
Isabelle BLOCH and Henri MAÎTRE

3.1. Objectives of fusion in image processing 47

3.2. Fusion situations 50

3.3. Data characteristics in image fusion 51

3.4. Constraints 54

3.5. Numerical and symbolic aspects in image fusion 55

3.6. Bibliography 56

Chapter 4. Fusion in Robotics 57
Michèle ROMBAUT

4.1. The necessity for fusion in robotics 57

4.2. Specific features of fusion in robotics 58

4.2.1.Constraints on the perception system 58

4.2.2. Proprioceptive and exteroceptive sensors 58

4.2.3. Interaction with the operator and symbolic interpretation 59

4.2.4. Time constraints 59

4.3. Characteristics of the data in robotics 61

4.3.1. Calibrating and changing the frame of reference 61

4.3.2. Types and levels of representation of the environment 62

4.4. Data fusion mechanisms 63

4.5. Bibliography 64

Chapter 5. Information and Knowledge Representation in Fusion Problems 65
Isabelle BLOCH and Henri MAÎTRE

5.1. Introduction 65

5.2. Processing information in fusion 65

5.3. Numerical representations of imperfect knowledge 67

5.4. Symbolic representation of imperfect knowledge 68

5.5. Knowledge-based systems 69

5.6. Reasoning modes and inference 73

5.7. Bibliography 74

Chapter 6. Probabilistic and Statistical Methods 77
Isabelle BLOCH, Jean-Pierre LE CADRE and Henri MAÎTRE

6.1. Introduction and general concepts 77

6.2. Information measurements 77

6.3. Modeling and estimation 79

6.4. Combination in a Bayesian framework 80

6.5. Combination as an estimation problem 80

6.6. Decision 81

6.7. Other methods in detection 81

6.8. An example of Bayesian fusion in satellite imagery 82

6.9. Probabilistic fusion methods applied to target motion analysis 84

6.9.1. General presentation 84

6.9.2. Multi-platform target motion analysis 95

6.9.3. Target motion analysis by fusion of active and passive measurements 96

6.9.4. Detection of a moving target in a network of sensors 98

6.10. Discussion 101

6.11. Bibliography 104

Chapter 7. Belief Function Theory 107
Isabelle BLOCH

7.1. General concept and philosophy of the theory 107

7.2. Modeling 108

7.3. Estimation of mass functions 111

7.3.1. Modification of probabilistic models 112

7.3.2. Modification of distance models 114

7.3.3. A priori information on composite focal elements (disjunctions) 114

7.3.4. Learning composite focal elements 115

7.3.5. Introducing disjunctions by mathematical morphology 115

7.4. Conjunctive combination 116

7.4.1. Dempster’s rule 116

7.4.2. Conflict and normalization 116

7.4.3. Properties 118

7.4.4. Discounting 120

7.4.5. Conditioning 120

7.4.6. Separable mass functions 121

7.4.7. Complexity 122

7.5. Other combination modes 122

7.6. Decision 122

7.7. Application example in medical imaging 124

7.8. Bibliography 131

Chapter 8. Fuzzy Sets and Possibility Theory 135
Isabelle BLOCH

8.1. Introduction and general concepts 135

8.2. Definitions of the fundamental concepts of fuzzy sets 136

8.2.1. Fuzzy sets 136

8.2.2. Set operations: Zadeh’s original definitions 137

8.2.3. α-cuts 139

8.2.4. Cardinality 139

8.2.5. Fuzzy number 140

8.3. Fuzzy measures 142

8.3.1. Fuzzy measure of a crisp set 142

8.3.2. Examples of fuzzy measures 142

8.3.3. Fuzzy integrals 143

8.3.4. Fuzzy set measures 145

8.3.5. Measures of fuzziness 145

8.4. Elements of possibility theory 147

8.4.1. Necessity and possibility 147

8.4.2. Possibility distribution 148

8.4.3. Semantics 150

8.4.4. Similarities with the probabilistic, statistical and belief interpretations 150

8.5. Combination operators 151

8.5.1. Fuzzy complementation 152

8.5.2. Triangular norms and conorms 153

8.5.3. Mean operators 161

8.5.4. Symmetric sums 165

8.5.5. Adaptive operators 167

8.6. Linguistic variables 170

8.6.1. Definition 171

8.6.2. An example of a linguistic variable 171

8.6.3. Modifiers 172

8.7. Fuzzy and possibilistic logic 172

8.7.1. Fuzzy logic 173

8.7.2. Possibilistic logic 177

8.8. Fuzzy modeling in fusion 179

8.9. Defining membership functions or possibility distributions 180

8.10. Combining and choosing the operators 182

8.11. Decision 187

8.12. Application examples 188

8.12.1. Example in satellite imagery 188

8.12.2. Example in medical imaging 192

8.13. Bibliography 194

Chapter 9. Spatial Information in Fusion Methods 199
Isabelle BLOCH

9.1. Modeling 199

9.2. The decision level 200

9.3. The combination level 201

9.4. Application examples 201

9.4.1. The combination level: multi-source Markovian classification 201

9.4.2. The modeling and decision level: fusion of structure detectors using belief function theory 202

9.4.3. The modeling level: fuzzy fusion of spatial relations 205

9.5. Bibliography 211

Chapter 10. Multi-Agent Methods: An Example of an Architecture and its Application for the Detection, Recognition and Identification of Targets 213
Fabienne EALET, Bertrand COLLIN and Catherine GARBAY

10.1.The DRI function 214

10.1.1. The application context 215

10.1.2. Design constraints and concepts 216

10.1.3. State of the art 216

10.2. Proposed method: towards a vision system 217

10.2.1. Representation space and situated agents 218

10.2.2. Focusing and adapting 219

10.2.3. Distribution and co-operation 220

10.2.4. Decision and uncertainty management 221

10.2.5. Incrementality and learning 221

10.3. The multi-agent system: platform and architecture 222

10.3.1. The developed multi-agent architecture 222

10.3.2. Presentation of the platformused 222

10.4. The control scheme 224

10.4.1. The intra-image control cycle 224

10.4.2. Inter-image control cycle 226

10.5. The information handled by the agents 227

10.5.1. The knowledge base 227

10.5.2. The world model 229

10.6. The results 231

10.6.1. Direct analysis 232

10.6.2. Indirect analysis: two focusing strategies 235

10.6.3. Indirect analysis: spatial and temporal exploration 237

10.6.4. Conclusion 240

10.7. Bibliography 241

Chapter 11. Fusion of Non-Simultaneous Elements of Information: Temporal Fusion 245
Michèle ROMBAUT

11.1. Time variable observations 245

11.2. Temporal constraints 246

11.3. Fusion 247

11.3.1. Fusion of distinct sources 247

11.3.2. Fusion of single source data 248

11.3.3. Temporal registration 249

11.4. Dating measurements 249

11.5. Evolutionary models 250

11.6. Single sensor prediction-combination 252

11.7. Multi-sensor prediction-combination 253

11.8. Conclusion 257

11.9. Bibliography 257

Chapter 12. Conclusion 259
Isabelle BLOCH

12.1. A few achievements 259

12.2. A few prospects 260

12.3. Bibliography 261

Appendices 263

A. Probabilities: A Historical Perspective 263

A.1. Probabilities through history 264

A.1.1. Before 1660 264

A.1.2. Towards the Bayesian mathematical formulation 266

A.1.3. The predominance of the frequentist approach: the “objectivists” 268

A.1.4. The 20th century: a return to subjectivism 269

A.2. Objectivist and subjectivist probability classes 271

A.3. Fundamental postulates for an inductive logic 272

A.3.1. Fundamental postulates 273

A.3.2. First functional equation 274

A.3.3. Second functional equation 275

A.3.4. Probabilities inferred from functional equations 276

A.3.5. Measure of uncertainty and information theory 276

A.3.6. De Finetti and betting theory 277

A.4.Bibliography 280

B. Axiomatic Inference of the Dempster-Shafer Combination Rule 283

B.1. Smets’s axioms 284

B.2. Inference of the combination rule 286

B.3.RelationwithCox’s postulates 287

B.4.Bibliography 289

List of Authors 291

Index 293